Most phenomena in the scientific field and other domains can be described and classified
as nonlinear diffusion equation which normally results from natural phenomena
that appear in our daily lives such as water waves at the beach caused by wind or tides, the
movement of a ship, or by raindrops; the same applies to other physical and mathematical
phenomena. In this study, we tried to find a solution to this kind of equations.
IN CHAPTER ONE, we gave brief history of the beginning of the study of waves and we
talked about some famous scientists who were interested in this field.
IN CHAPTER TWO, we highlighted the diversity and classification of equations in terms of:
Linear, Non-linear, Dispersive and Non-dispersive.
IN CHAPTER THREE, we introduced the Painlevé method and we applied it into the KdV and
modified KdV equations, and in addition to that, we were able to find analytic solutions
for these equations.
IN CHAPTERS FOUR AND FIVE, we showed several methods of scheme difference, we

focused
our study on the non-linear term of the KdV equation.
IN CHAPTER SIX, we gave some examples of the scheme difference methods and we applied
them by Matlab programs. Moreover, our work is supported by pictures and figures.
CHAPTER SEVEN shows the future works, we enhanced the work by Appendix.